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Power Transformation [To Gaussian] {Box-Cox}

Description

This technique is similar to log transformation but allows for a broader range of transformations. The most common power transformation is the Box-Cox transformation, which raises the feature values to a power that is determined using maximum likelihood estimation.

Formula

\[ x_{transformed} = \frac{x^{\lambda} - 1}{\lambda} \]

Here, \(x\) is the original feature value, and \(\lambda\) is the power parameter that is estimated using maximum likelihood.

Varieties

The Box-Cox transform is another popular function belonging to the power transform family of functions. This function has a prerequisite that the numeric values to be transformed must be positive (similar to what log transform expects). In case they are negative, shifting using a constant value helps.

\[ y_i^{(\lambda)} = \begin{cases} \frac{y_i^{\lambda} - 1}{\lambda} & \text{if } \lambda \neq 0, \\ \ln y_i & \text{if } \lambda = 0, \end{cases} \]

The resulting transformed output y is a function of input x and the transformation parameter \(\lambda\) such that when \(\lambda\) = 0, the resultant transform is the natural log transform which we discussed earlier. The optimal value of \(\lambda\) is usually determined using a maximum likelihood or log-likelihood estimation.

Example

Let's now apply the Box-Cox transform on our developer income feature. First we get the optimal lambda value from the data distribution by removing the non-null values as follows.

income = np.array(fcc_survey_df["Income"])
income_clean = income[~np.isnan(income)]
l, opt_lambda = spstats.boxcox(income_clean)
print("Optimal lambda value:", opt_lambda)

# Optimal lambda value: 0.117991239456

Now that we have obtained the optimal \(\lambda\) value, let us use the Box-Cox transform for two values of \(\lambda\) such that \(\lambda = 0\) and \(\lambda = \lambda_{optimal}\) and transform the developer Income feature.

fcc_survey_df["Income_boxcox_lambda_0"] = spstats.boxcox(
    (1 + fcc_survey_df["Income"]),
    lmbda=0
)

fcc_survey_df["Income_boxcox_lambda_opt"] = spstats.boxcox(
    fcc_survey_df["Income"],
    lmbda=opt_lambda
)

Info

The transformed features are depicted in the above data frame. Just like we expected, Income_log and Income_boxcox_lamba_0 have the same values.

The distribution looks more normal-like similar to what we obtained after the log transform.